Theorem nfcnv 5835

Description: Bound-variable hypothesis builder for converse relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypothesis

Ref	Expression
nfcnv.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfcnv	⊢ Ⅎ𝑥◡𝐴

Proof of Theorem nfcnv

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cnv 5640	. 2 ⊢ ◡𝐴 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
2		nfcv 2899	. . . 4 ⊢ Ⅎ𝑥𝑧
3		nfcnv.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfcv 2899	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 5147	. . 3 ⊢ Ⅎ𝑥 𝑧𝐴𝑦
6	5	nfopab 5169	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
7	1, 6	nfcxfr 2897	1 ⊢ Ⅎ𝑥◡𝐴

Syntax hints:  Ⅎwnfc 2884   class class class wbr 5100  {copab 5162  ^◡ccnv 5631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-cnv 5640

This theorem is referenced by:  nfrn  5909  nfpred  6272  nffun  6523  nff1  6736  funcnvmpt  6951  nfsup  9366  nfinf  9398  gsumcom2  19916  ptbasfi  23537  mbfposr  25621  itg1climres  25683  nfwsuc  36032  aomclem8  43418  rfcnpre1  45379  rfcnpre2  45391  smfpimcc  47166